Path-Complete and Neural Lyapunov Functions: Computation and Performance
(2024)
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- Authors
- Supervisors
- Jungers, Raphaël
- Abstract
- Switched systems are essential in modern engineering due to their ability to model complex systems with transitions between different modes of operation. Their stability poses significant challenges because of the interplay between discrete switching and dynamics, requiring advanced mathematical tools for analysis. While Lyapunov theory is widely used to prove stability, classical methods often struggle with the added complexity of switched systems. This has led to research on extending Lyapunov theory to better address these challenges. The introduction of path-complete Lyapunov functions brought a new perspective by incorporating combinatorial structures to encode the switching signals of the switched system. This thesis extends the study of path-complete Lyapunov functions by addressing the template-dependent ordering of graphs, i.e., comparing stability certificates while considering specific classes of Lyapunov functions. We introduce template-dependent lifts. These are combinatorial operations on graphs, that characterize the ordering of graphs concerning templates that share a common closure property, such as addition or minimum. This novel approach enhances the understanding of conservatism in stability conditions and guides the selection of graph-template pairs for stability analysis. Additionally, we explore neural Lyapunov functions as a modern approach to approximating the joint spectral radius (JSR) of linear switched systems. We present a framework that fine-tunes neural networks to approximate the JSR with theoretical and empirical guarantees of effectiveness. We leverage machine learning techniques and the CEGIS approach to provide formal correctness in neural Lyapunov functions, demonstrating promising results against classical methods.
- Affiliations
UCLouvain SST/ICTM/INMA - Pôle en ingénierie mathématique
Citations
Debauche, V. (2024). Path-Complete and Neural Lyapunov Functions: Computation and Performance. https://hdl.handle.net/2078.5/233158